The Mishnah on today’s daf (=page) mentions a number of things that were forbidden by the Sages if a Jewish person was not involved in its production. For example, when an animal is milked without a Jewish person watching, or when bread is baked by a non-Jew – even if we are certain that there are no non-kosher ingredients – it is forbidden.
Regarding milk, the Gemara asks why there should be any concern since there is a distinctive difference in the color of kosher and non-kosher milk that we can recognize. The Gemara concludes that there may be small amounts of non-kosher ingredients added that are not obvious. Although the ruling of the Rambam and the Shulhan Arukh is that milk is only kosher if a Jew supervises the milking, given that the only concern is that there may be forbidden additives placed in the milk, there are contemporary responsa that permit the use of milk that has reliable government supervision.
The concern with bread is not the ingredients – which must be kosher – but the possibility that joining with non-Jews at meals may lead to intermarriage and assimilation. The Gemara itself suggests that there may be situations where the bread is commercially baked where no such concern exists. Nevertheless, the simple reading of the Gemara seems to indicate that the Rabbinic injunction against non-Jewish bread that is brought in the Mishnah could not be removed.
The Rambam, however, rules that in many places where bread baked by Jews was not readily available, people traditionally purchased commercially baked bread from non-Jews. The Rema attests to the fact that in the Ashkenazi communities it was commonplace to purchase such bread even if bread baked by Jews was available. This ruling is based on the fact that many rishonim (e.g. Tosafot and the Mordekhai) believe that this rabbinic injunction never became widespread and was not accepted by the masses, so there was no need to formally rescind it.